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MATH 311–505/506 Fall 2009 Sample problems for the final exam Any problem may be altered or replaced by a different one! Problem 1 (15 pts.) Find a quadratic polynomial p(x) = ax2 + bx + c such that p(−1) = p(3) = 6 and p′ (2) = p(1). Problem 2 (20 pts.) Let v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 1). Let L : R3 → R3 be a linear operator on R3 such that L(v1 ) = v2 , L(v2 ) = v3 , L(v3 ) = v1 . (i) Show that the vectors v1 , v2 , v3 form a basis for R3 . (ii) Find the matrix of the operator L relative to the basis v1 , v2 , v3 . (iii) Find the matrix of the operator L relative to the standard basis. 1 1 1 Problem 3 (20 pts.) Let B = 1 1 1 . 1 1 1 (i) Find all eigenvalues of the matrix B. (ii) Find a basis for R3 consisting of eigenvectors of B? (iii) Find an orthonormal basis for R3 consisting of eigenvectors of B? Problem 4 (20 pts.) Find a quadratic polynomial q that is the best least squares fit to the function f (x) = |x| on the interval [−1, 1]. This means that q should minimize the distance Z 1 1/2 2 dist(f, q) = |f (x) − q(x)| dx −1 over all polynomials of degree at most 2. Problem 5 (25 pts.) It is known that 2 Z x 2x 2 2 x sin(ax) dx = − + 3 cos(ax) + 2 sin(ax) + C, a a a a 6= 0. (i) Find the Fourier sine series of the function f (x) = x2 on the interval [0, π]. (ii) Over the interval [−3.5π, 3.5π], sketch the function to which the series converges. (iii) Describe how the answer to (ii) would change if we studied the Fourier cosine series instead. Bonus Problem 6 (15 pts.) equation ∂u ∂2u = ∂t ∂x2 Solve the initial-boundary value problem for the heat (0 < x < π, t > 0), u(x, 0) = 1 + 2 cos(2x) − cos(3x) ∂u ∂u (0, t) = (π, t) = 0 ∂x ∂x (t > 0). (0 < x < π),